Theorem eldmcoss 39087

Description: Elementhood in the domain of cosets. (Contributed by Peter Mazsa, 29-Mar-2019.)

Assertion

Ref	Expression
eldmcoss	⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom ≀ 𝑅 ↔ ∃𝑢 𝑢𝑅𝐴))

Distinct variable groups:   𝑢,𝐴   𝑢,𝑅   𝑢,𝑉

Proof of Theorem eldmcoss

Step	Hyp	Ref	Expression
1		dmcoss3 39082	. . 3 ⊢ dom ≀ 𝑅 = dom ◡𝑅
2	1	eleq2i 2861	. 2 ⊢ (𝐴 ∈ dom ≀ 𝑅 ↔ 𝐴 ∈ dom ◡𝑅)
3		eldmcnv 38884	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom ◡𝑅 ↔ ∃𝑢 𝑢𝑅𝐴))
4	2, 3	bitrid 286	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom ≀ 𝑅 ↔ ∃𝑢 𝑢𝑅𝐴))

Syntax hints:   → wi 4   ↔ wb 209  ∃wex 1806   ∈ wcel 2149   class class class wbr 5113  ^◡ccnv 5661  dom cdm 5662   ≀ ccoss 38722

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5114  df-opab 5178  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-coss 39040

This theorem is referenced by:  eldmcoss2  39088  eldm1cossres  39089